11 THE SPECTRAL THEORY OF HUBERT 411 (b) With the notation of Problem 15, let C be the operator of rank 1 defined by C - x=:-(x | e_i)e0 • Show that Sp(t/4- C) is the disk |£| ^ 1 while Sp(C7) is the circle |£| = 1. (Consider separately the restrictions of U + C to the subspace generated by the en with n ^ 0 and to its orthogonal supplement.) (c) Let N be a normal operator on E and C a compact operator on E. If Sp(AO is nondenumerable, show that the same is true of Sp((W 4- C)*(N -f C)) (use (a) above and (15.11.8(i)). Deduce that the one-sided shift operator V (Problem 19(a)) cannot be of the form N+C (observe that K*F= 1B). 21. Let E be an infinite-dimensional separable Hilbert space. (a) Show that every nonzero two-sided ideal $ of the ring & (E) contains the ideal <5 of operators of finite rank. (If T ¥= 0 belongs to $, show that every operator of rank 1 can be written in the form BTC for suitably chosen operators B and C.) (b) Show that the only closed two-sided ideal of the Banach algebra -S?(E), other than -$?(E) and {0}, is the ideal & of compact operators. (First observe that <£ is the closure of @, and then that if a two-sided ideal contains a noncompact operator, then it also contains a noncompact positive hermitian operator H (Problem 6). For such an operator, show that there exists an interval M = [a, H- oo [ with a > 0 such that the space E(M) (in the notation of (15.11.9)) is infinite-dimensional. If Vis an isometry of E onto E(M), show that V*HVi& invertible.) 22. Let E be an infinite-dimensional separable Hilbert space. An operator with index on E is a continuous operator T such that (1) T(E) is closed and of finite codimension; (2) r-1(0) is finite-dimensional. (a) If T is an index operator, show that there exists a continuous operator A such that 1E — A Tand 1E — TA are of finite rank. (Show that Tis a homeomorphism of the orthogonal supplement F of J^O) onto T(B)9 and take A to be the inverse homeo- morphism on F and zero on the orthogonal supplement of F.) (b) Conversely, suppose that Tis a continuous operator on E, for which there exists a continuous operator A such that 1E — AT and 1E — TA are compact. Show that T is an index operator, (Using (11.3.2), show first that the kernels of T and T* are finite- dimensional, and hence that T(E) has finite codimension. Then use the fact that the restriction of AT to the orthogonal supplement F of the kernel of AT is a homeo- morphism onto its image, and finally use (12.13.2(iii)).) 23. Let E be an infinite-dimensional separable Hilbert space. (a) Let T be a continuous operator on E such that T~l(Q) is of infinite dimension. Then E is the Hilbert sum of an infinite sequence (En)n>0 of infinite-dimensional sub- spaces, such that the Ert with n S> 1 are contained in 71" *(()). For each n ^ 1, let Sn denote an isometry of E0 onto En. Let A be the continuous operator which on E0 is equal to Si, and on En is equal toSn + iSnl9 for all n ^ 1. Also let Fbe the operator which is zero on E0, is equal to Srl on EI, and is equal to Sn-j.Snl on £„ for all n i> 2. Let TQ denote the restriction ofPEQTto E0, and let B be the continuous oper- ator which is equal to VT on E0, to — ToSj;1 on EI, and to —Sn,iT0Snl onEnfor all n i> 2. Prove that T = AB — BA. (b) Deduce from (a) that for any continuous operator T7 on E there exist four operators A, B, C, D such that T = (AB - BA) + (CD - DC). (Write T as the sum of two continuous operators, each of which has an infinite- dimensional kernel.)s that -1 is an eigenvalue of T~1C.)uch that (jitw) converges vaguely to a measure which does not belong to M£(X, v).